How dark matter halos get their cusps

How do the density profiles of dark matter halos emerge from their hierarchical assembly? We address this question using the Boltzmann–Poisson equations for the evolution of the coarse-grained distribution function of a halo composed of subhalos with a continuous mass spectrum in an expanding universe. We obtain an exact self-similar solution that connects the halo density log-slope γ, defined by ρ ∝ r⁻ᵞ, to its mass growth M ∝ tˢ. Accounting for relaxation through gravitational scattering and capture of subhalos, we find γ = 6/(s + 2).

 

This framework links three characteristic density profiles to distinct stages of halo assembly. During the initial collapse, Keplerian infall and a stationary density amplitude select s = 2, producing the prompt cusp with a density log-slope of γ = 3/2. During hierarchical accretion, the halo mass growth is set by the spectral index n of the linear power spectrum and the linear growth factor D ∝ tᵖ, giving s = 6p/(3 + n). In the matter-dominated Einstein–de Sitter era, the resulting density log-slope follows the Syer–White relation, γ = 3(3 + n)/(5 + n). We argue that stable clustering selects an attractive fixed point with spectral index n = −2, yielding γ = 1, the inner cusp of the Navarro–Frenk–White (NFW) profile. Finally, when accretion stalls, as may occur during dark energy domination, we have s = 0 and the solution approaches the outer NFW log-slope of γ = 3. Together, these results provide a dynamical framework for understanding the origin of halo profiles in the non-linear regime.